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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n325 at Knotilus!

Link Presentations

[edit Notes on L11n325's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X4,17,1,18 X21,11,22,10
Gauss code {1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11, -7, 9}, {-9, 2, 4, -5, 10, 6, -8, 7, -11, 8}
A Braid Representative
A Morse Link Presentation L11n325 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1)^2 (w-1)^2}{\sqrt{u} v w} (db)
Jones polynomial - q^{-8} +3 q^{-7} -6 q^{-6} +9 q^{-5} -10 q^{-4} +12 q^{-3} -9 q^{-2} +2 q+8 q^{-1} -4 (db)
Signature -2 (db)
HOMFLY-PT polynomial -z^4 a^6-2 z^2 a^6-2 a^6+z^6 a^4+4 z^4 a^4+8 z^2 a^4+a^4 z^{-2} +6 a^4-3 z^4 a^2-8 z^2 a^2-2 a^2 z^{-2} -7 a^2+2 z^2+ z^{-2} +3 (db)
Kauffman polynomial z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+4 z^7 a^7-6 z^5 a^7+z a^7+3 z^8 a^6-z^6 a^6-6 z^4 a^6+4 z^2 a^6+z^9 a^5+5 z^7 a^5-11 z^5 a^5+8 z^3 a^5-3 z a^5+5 z^8 a^4-8 z^6 a^4+10 z^4 a^4-11 z^2 a^4-a^4 z^{-2} +7 a^4+z^9 a^3+2 z^7 a^3-3 z^5 a^3+5 z^3 a^3-6 z a^3+2 a^3 z^{-1} +2 z^8 a^2-4 z^6 a^2+13 z^4 a^2-18 z^2 a^2-2 a^2 z^{-2} +9 a^2+z^7 a+z^5 a-z^3 a-3 z a+2 a z^{-1} +3 z^4-6 z^2- z^{-2} +4 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
3         22
1        2 -2
-1       62 4
-3      54  -1
-5     74   3
-7    57    2
-9   45     -1
-11  25      3
-13 14       -3
-15 2        2
-171         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}_2^{2} {\mathbb Z}^{2}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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See/edit the Link Page master template (intermediate).

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